Valdez Kirk
07/30/2023 · Middle School
Use the change-of-base formula and a calculator to evaluate the logarithm. Round your answer to three decimal places. \( \log _{1 / 6} 5 \) \( \log _{1 / 6} 5 \approx \square \) (Do not round until the final answer. Then round to three decimal places as needed.)
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Step-by-step Solution
为了求解 \(\log_{1/6} 5\)，我们可以使用换底公式。换底公式如下：
\[
\log_a b = \frac{\log_c b}{\log_c a}
\]
在这个问题中，我们需要计算 \(\log_{1/6} 5\)，所以我们可以选择以10为底的对数（常用对数）来计算。具体步骤如下：
1. **确定换底公式**：
\[
\log_{1/6} 5 = \frac{\log 5}{\log (1/6)}
\]
2. **计算 \(\log 5\)**：
使用计算器，我们得到：
\[
\log 5 \approx 0.6990
\]
3. **计算 \(\log (1/6)\)**：
注意到 \(\log (1/6) = \log 1 - \log 6\)，而 \(\log 1 = 0\)，所以：
\[
\log (1/6) = 0 - \log 6 = -\log 6
\]
使用计算器，我们得到：
\[
\log 6 \approx 0.7782
\]
因此：
\[
\log (1/6) = -0.7782
\]
4. **将结果代入换底公式**：
\[
\log_{1/6} 5 = \frac{\log 5}{\log (1/6)} = \frac{0.6990}{-0.7782}
\]
5. **计算最终结果**：
使用计算器，我们得到：
\[
\frac{0.6990}{-0.7782} \approx -0.898
\]
6. **四舍五入到小数点后三位**：
最终结果为：
\[
\log_{1/6} 5 \approx -0.898
\]
因此，\(\log_{1/6} 5 \approx -0.898\)。
Quick Answer
\(\log_{1/6} 5 \approx -0.898\)
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